Umegaki relative entropy of local recovery
- Field
- Topics
Problem
Does the conditional mutual information of every finite-dimensional tripartite state dominate its Umegaki relative entropy of local recovery? For a state \(\rho_{ABC}\), define
With \(D(\tau\Vert\omega):=\operatorname{Tr}[\tau(\log\tau-\log\omega)]\) when \(\operatorname{supp}\tau\subseteq\operatorname{supp}\omega\), the question is whether the quantity in Eq. (1) always satisfies
where the minimum in Eq. (2) is over all completely positive trace-preserving recovery maps \(\mathcal R_{B\to BC}\).
Source
The local Umegaki-recovery inequality was formulated by Li and Winter [LW18]; the associated universal recovery-map proposal is recorded as Eq. (12.153) of Wilde’s text [Wil17].
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Li and Winter formulated the local Umegaki-recovery inequality in Eq. (2) and related it to a proposed universal, functorial recovery-map strengthening of relative-entropy monotonicity [LW18]. Wilde recorded the latter proposal as Eq. (12.153) in his text [Wil17].
Fawzi and Fawzi disproved Eq. (2). Their explicit family consists of the pure three-qubit states
\begin{equation} \begin{aligned} \rho^{(\theta)}_{ABC} &=\lvert\psi_\theta\rangle\!\langle\psi_\theta\rvert,\\ \lvert\psi_\theta\rangle_{ABC} &=\frac{1}{\sqrt2}\lvert0\rangle_B\lvert00\rangle_{AC} +\frac{1}{\sqrt2}\lvert1\rangle_B \bigl(\cos\theta\,\lvert01\rangle_{AC} +\sin\theta\,\lvert10\rangle_{AC}\bigr). \end{aligned} \tag{3} \end{equation}For sufficiently small positive \(\theta\), the states in Eq. (3) violate Eq. (2). The violation is certified by optimizing a semidefinite-representable Petz–Rényi-divergence lower bound on the relative entropy of recovery [FF18].
Comment
The answer to the local-recovery question in Eq. (2) is negative. This is the conditional-mutual- information consequence of the recovery proposal associated with Eq. (12.153) of [Wil17]. The archived statement is deliberately restricted to local recovery: it does not assert that the same counterexample rules out every nonfunctorial recovery channel allowed to act jointly on an otherwise spectator system.